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On regular sets in Cayley graphs

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Abstract

Let \(\Gamma = (V, E)\) be a graph and a, b nonnegative integers. An (a, b)-regular set in \(\Gamma \) is a nonempty proper subset D of V such that every vertex in D has exactly a neighbours in D and every vertex in \(V{\setminus }D\) has exactly b neighbours in D. A (0, 1)-regular set is called a perfect code, an efficient dominating set, or an independent perfect dominating set. A subset D of a group G is called an (a, b)-regular set of G if it is an (a, b)-regular set in some Cayley graph of G, and an (a, b)-regular set in a Cayley graph of G is called a subgroup (a, b)-regular set if it is also a subgroup of G. In this paper, we study (a, b)-regular sets in Cayley graphs with a focus on (0, k)-regular sets, where \(k \ge 1\) is an integer. Among other things, we determine when a non-trivial proper normal subgroup of a group is a (0, k)-regular set of the group. We also determine all subgroup (0, k)-regular sets of dihedral groups and generalized quaternion groups. We obtain necessary and sufficient conditions for a hypercube or the Cartesian product of n copies of the cycle of length p to admit (0, k)-regular sets, where p is an odd prime. Our results generalize several known results from perfect codes to (0, k)-regular sets.

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Acknowledgements

The authors would like to thank the anonymous reviews for their valuable comments which improved the paper. Xu was supported by the National Natural Science Foundation of China (Grant No.@ 12071194, 11901263).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University, Lanzhou, 730000, Gansu, China

    Xiaomeng Wang & Shou-Jun Xu

  2. School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Sanming Zhou

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  1. Xiaomeng Wang
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Correspondence to Shou-Jun Xu.

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Wang, X., Xu, SJ. & Zhou, S. On regular sets in Cayley graphs. J Algebr Comb 59, 735–759 (2024). https://doi.org/10.1007/s10801-024-01298-y

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  • DOI: https://doi.org/10.1007/s10801-024-01298-y

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